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UNIT II: The Basic Theory

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UNIT II: The Basic Theory

- Zero-sum Games
- Nonzero-sum Games
- Nash Equilibrium: Properties and Problems
- Bargaining Games
- Review
- Midterm 3/23

3/9

Review

- Review Terms
- Counting Strategies
- Prudent v. Best-Response Strategies
- Graduate Assignment

Review

Dominant Strategy: A strategy that is best no matter what the opponent(s) choose(s).

Prudent Strategy:A prudent strategy maximizes the minimum payoff a player can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i.

Mixed Strategy:A mixed strategy for player i is a probability distribution over all strategies available to player i.

Best Response Str’gy: A strategy, s’, is a best response strategy iff Pi(s’,t) > Pi(s,t) for all s.

Dominated Strategy: A strategy is dominated if it is never a best response strategy.

Review

Saddlepoint:A set of prudent strategies (one for each player), s. t. (s*, t*) is a saddlepoint, iff maxmin = minmax.

Nash Equilibrium: a set of best response strategies (one for each player), (s’,t’) such that s’ is a best response to t’ and t’ is a b.r. to s’.

Subgame: a part (or subset) of an extensive game, starting at a singleton node (not the initial node) and continuing to payoffs.

Subgame Perfect Nash Equilibrium (SPNE): a NE achieved by strategies that also constitute NE in each subgame. eliminates NE in which the players threats are not credible.

Counting Strategies

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-2 4 2 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Counting Strategies

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-2 4 2 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Counting Strategies

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-2 4 2 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Counting Strategies

Player 1

Player 2 has 4 strategies:

Left Right

L R L R

-242 -1

Player 2

LL RR LR RL

-2 4 -2 4

2 -1 -1 2

L

R

GAME 2: Button-Button

Counting Strategies

Player 1

If Player 2 cannot observe Player 1’s choice …

Player 2 will have fewer strategies.

Left Right

L R L R

-2 4 2 -1

Player 2

GAME 2: Button-Button

Counting Strategies

L R

L

R

Player 1

-2 4

2 -1

Left Right

L R L R

-2 4 2 -1

Player 2

GAME 2: Button-Button

Prudence v. Best Response

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 2

Find all the NE of the game.

Prudence v. Best Response

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 2

NE = {(1, 1); (0, 0); }

Prudence v. Best Response

Battle of the Sexes

O F

O

F

Player 1

2, 1 0, 0

0, 0 1, 2

Opera Fight

O F O F

(2,1) (0,0) (0,0) (1,2)

Player 2

NE = {(O,O); (F,F); }

Battle of the Sexes

Prudence v. Best Response

O F

P2

2

1

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

1 2

P1

NE = {(1, 1); (0, 0); (MNE)}

Mixed Nash Equilibrium

Battle of the Sexes

Prudence v. Best Response

OF

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (MNE)}

Battle of the Sexes

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olq) = 2q

EP1(Flq) = 1-1q

EP2(Olp) = 1p

EP2(Flp)= 2-2p

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (MNE)}

Battle of the Sexes

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olq) = 2q

EP1(Flq) = 1-1q

q* = 1/3

EP2(Olp) = 1p

EP2(Flp)= 2-2p

p* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (2/3,1/3)}

}

Battle of the Sexes

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

2/3

0

2

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

2/3

0

2

EP1 = 2q +0(1-q)

2,1 0,0

0, 0 1, 2

p=1

p=0

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

p=0

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

p=1

2,1 0,0

0, 0 1, 2

p=1

p=0

EP1 = 0q+1(1-q)

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

Opera

2,1 0,0

0, 0 1, 2

p=1

p=0

Fight

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

q=1 q=0

Player 1’s Expected Payoff

EP1

1

2/3

0

2

0

2,1 0,0

0, 0 1, 2

p=1

p=0

p=1

0<p<1

p=0

0<p<1

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

If Player 1 uses her (mixed) b-r strategy (p=2/3), her expected payoff varies from 1/3 to 4/3.

q=1 q=0

EP1

2/3

1/3

2

0

2,1 0,0

0, 0 1, 2

p = 2/3

p=1

p=0

4/3

p=1

p=0

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

If Player 2 uses his (mixed) b-r strategy (q=1/3), the expected payoff to Player 1 is 2/3, for all p.

q=1 q=0

EP1

2/3

1/3

2/3

2

0

2,1 0,0

0, 0 1, 2

p=1

p=0

4/3

p=1

p=0

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

O F

Find the prudent strategy for each player.

2, 1 0, 0

0, 0 1, 2

O

F

Battle of the Sexes

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olp) = 2p

EP1(Flp) = 1-1p

p* = 1/3

EP2(Oiq) = 1q

EP2(Flq)= 2-2q

q* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

Prudent strategies: 1/3; 2/3

Battle of the Sexes

Prudence v. Best Response

O F

Let (p,1-p) = prob1(O, F)

(q,1-q) = prob2(O, F)

Then

EP1(Olq) = 2q

EP1(Flq) = 1-1q

q* = 1/3

EP2(Olp) = 1p

EP2(Flp)= 2-2p

p* = 2/3

2, 1 0, 0

0, 0 1, 2

O

F

NE = {(1, 1); (0, 0); (2/3,1/3)}

Battle of the Sexes

Prudence v. Best Response

If Player 1 uses her prudent strategy (p=1/3), expected payoff is 2/3 no matter what player 2 does

O F

EP1

2/3

1/3

2

0

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Battle of the Sexes

Prudence v. Best Response

If both players use (mixed) b-r strategies, expected payoff is 2/3 for each.

O F

EP1

2/3

1/3

2

0

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q = 1/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Battle of the Sexes

Prudence v. Best Response

If both players use (mixed) b-r strategies, expected payoff is 2/3 for each.

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Battle of the Sexes

Prudence v. Best Response

If both players use prudent strategies, expected payoff is 2/3 for each.

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

Battle of the Sexes

Prudence v. Best Response

BATNA: Best Alternative to a Negotiated Agreement

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

BATNA

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent: {(1/3, 2/3)}

Battle of the Sexes

Prudence v. Best Response

O F

P2

2

1

2/3

2, 1 0, 0

0, 0 1, 2

NE (0,0)

O

F

NE (1,1)

2/31 2

P1

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Is the pair of prudent strategies an equilibrium?

Battle of the Sexes

Prudence v. Best Response

NO: Player 1’s best response to Player 2’s prudent strategy (q=2/3) is Opera (p=1).

O F

EP1

2/3

1/3

2/3

2

0

Opera

2, 1 0, 0

0, 0 1, 2

p = 2/3

O

F

4/3

p=1

p=0

p = 1/3

q = 1/32/3

q

NE = {(1, 1); (0, 0); (2/3,1/3)} Prudent:{(1/3, 2/3)}

Review

[I]f game theory is to provide a […] solution to a game-theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s predicted strategy must be that player’s best response to the predicted strategies of the other players. Such a prediction could be called strategically stable or self-enforcing, because no single player wants to deviate from his or her predicted strategy (Gibbons: 8).

Review

SADDLEPOINT v. NASH EQUILIBRIUM

STABILITY: Is it self-enforcing?

YES YES

UNIQUENESS: Does it identify an unambiguous course of action?

YES NO

EFFICIENCY: Is it at least as good as any other outcome for all players?

--- (YES) NOT ALWAYS

SECURITY: Does it ensure a minimum payoff?

YES NO

EXISTENCE: Does a solution always exist for the class of games? YES YES

Review

Problems of Nash Equilibrium

- Indeterminacy: Nash equilibria are not usually unique.
2. Inefficiency: Even when they are unique, NE are not always efficient.

Review

Problems of Nash Equilibrium

- T1 T2

Multiple and Inefficient Nash Equilibria

S1

S2

5,5 0,1

1,0 3,3

When is it advisable to play a prudent strategy in a nonzero-sum game?

Review

Problems of Nash Equilibrium

- T1 T2

Multiple and Inefficient Nash Equilibria

S1

S2

5,5 -99,1

1,-99 3,3

When is it advisable to play a prudent strategy in a nonzero-sum game?

What do we need to know/believe about the other player?

Bargaining Games

Bargaining games are fundamental to understanding the price determination mechanism in “small” markets.

The central issue in all bargaining games is credibility: the strategic use of threats, bluffs, and promises.

When information is asymmetric, profitable exchanges may be “left on the table.”

In such cases, there is an incentive to make oneself credible (e.g., appraisals; audits; “reputable” agents; brand names; lemons laws; “corporate governance”).

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